EEM478-WEEK8 Finite Impulse Response (FIR) Filters 478/icerik/eem478... · 2018. 4. 11. · Chapter 14, Slide 3 Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2002 Introduction

EEM478-WEEK8Finite Impulse Response (FIR) Filters

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2002Chapter 14, Slide 2

Learning Objectives

Introduction to the theory behind FIR filters: Properties (including aliasing). Coefficient calculation. Structure selection.

Implementation in Matlab, C, assembly and linear assembly.


Introduction

Amongst all the obvious advantages that digital filters offer, the FIR filter can guarantee linear phase characteristics.

Neither analogue or IIR filters can achieve this.

There are many commercially available software packages for filter design. However, without basic theoretical knowledge of the FIR filter, it will be difficult to use them.


Properties of an FIR Filter

Filter coefficients:

x[n] represents the filter input,bk represents the filter coefficients,y[n] represents the filter output,N is the number of filter coefficients

(order of the filter).




FIR equation

Filter structure




If the signal x[n] is replaced by an impulse [n] then:












Finally:




With:

The coefficients of a filter are the same as the impulse response samples of the filter.


Frequency Response of an FIR Filter

By taking the z-transform of h[n], H(z):

Replacing z by e-j in order to find the frequency response leads to:



Since e-j2k = 1 then:

Therefore:

FIR filters have a periodic frequency response and the period is 2.



Frequency response:

FIR y[n]x[n]

Fs/2Fs/2

Freq

Freq

x[n]

y[n]



Solution: Use an anti-aliasing filter.

FIR y[n]x[n]ADC

Analogue Anti-Aliasing

x(t)

Fs/2Fs/2FreqFreq

x(t)

y[n]


Phase Linearity of an FIR Filter

A causal FIR filter whose impulse response is symmetrical is guaranteed to have a linear phase response.

Even symmetry Odd symmetry



A causal FIR filter whose impulse response is symmetrical (ie h[n] = h[N-1-n] for n = 0, 1, …, N-1) is guaranteed to have a linear phase response.



Application of 90° linear phase shift:

Signal separation

90o

delay

90o

delay

+

+

+

-

I

Q

Reverse

Forward

IH

QH


Design Procedure

To fully design and implement a filter five steps are required:

(1) Filter specification.(2) Coefficient calculation.(3) Structure selection.(4) Simulation (optional).(5) Implementation.


Filter Specification - Step 1


Coefficient Calculation - Step 2

There are several different methods available, the most popular are: Window method. Frequency sampling. Parks-McClellan.

We will just consider the window method.


Window Method

First stage of this method is to calculate the coefficients of the ideal filter.

This is calculated as follows:


Window Method Second stage of this method is to select a window

function based on the passband or attenuation specifications, then determine the filter length based on the required width of the transition band.

Using the Hamming Window:


Window Method

The third stage is to calculate the set of truncated or windowed impulse response coefficients, h[n]:

for

Where:for


Window Method Matlab code for calculating coefficients:

close all;clear all;

fc = 8000/44100; % cut-off frequencyN = 133; % number of tapsn = -((N-1)/2):((N-1)/2);n = n+(n==0)*eps; % avoiding division by zero

[h] = sin(n*2*pi*fc)./(n*pi); % generate sequence of ideal coefficients[w] = 0.54 + 0.46*cos(2*pi*n/N); % generate window functiond = h.*w; % window the ideal coefficients

[g,f] = freqz(d,1,512,44100); % transform into frequency domain for plotting

figure(1)plot(f,20*log10(abs(g))); % plot transfer functionaxis([0 2*10^4 -70 10]);

figure(2);stem(d); % plot coefficient valuesxlabel('Coefficient number');ylabel ('Value');title('Truncated Impulse Response');

figure(3)freqz(d,1,512,44100); % use freqz to plot magnitude and phase responseaxis([0 2*10^4 -70 10]);


Window Method


Realisation Structure Selection - Step 3

Direct form structure for an FIR filter:



Linear phase structures:

N even:

N Odd:




(a) N even.(b) N odd.



Cascade structures:




Cascade structures:



Implementation - Step 5

Implementation procedure in ‘C’ with fixed-point: Set up the codec (\Links\CodecSetup.pdf).

Transform: to ‘C’ code.

(\Links\FIRFixed.pdf) Configure timer 1 to generate an interrupt at

8000Hz (\Links\TimerSetup.pdf). Set the interrupt generator to generate an

interrupt to invoke the Interrupt Service Routine (ISR) (\Links\InterruptSetup.pdf).



Implementation procedure in ‘C’ with floating-point:Same set up as fixed-point plus: Convert the input signal to floating-point

format. Convert the coefficients to floating-point

format. With floating-point multiplications there is

no need for the shift required when using Q15 format.

See \Links\FIRFloat.pdf



Implementation procedure in assembly:Same set up as fixed-point, however: is written in assembly.

(\Links\FIRFixedAsm.pdf)

The ISR is now declared as external.



Implementation procedure in assembly:The filter implementation in assembly is now using circular addressing and therefore: The circular pointers and block size register

are selected and initialised by setting the appropriate values of the AMR bit fields.

The data is now aligned using:

Set the initial value of the circular pointers, see \Links\FIRFixedAsm.pdf.

#pragma DATA_ALIGN (symbol, constant (bytes))



y0 = b0*x0 + b1*x1 + b2*x2 + b3*x3

Circular addressing link slide.

time

y[n]

0 1 2

*B5*B5*B5*B5

*A5*A5*A5*A5

b0b1b2b3

x0x1x2x3


x4x1x2x3


y0 = b0*x0 + b1*x1 + b2*x2 + b3*x3y1 = b0*x1 + b1*x2 + b3*x4+ b2*x3


time

y[n]

0 1 2

*B5*B5*B5*B5

*A5*A5*A5*A5

b0b1b2b3


x4x5x2x3


y0 = b0*x0 + b1*x1 + b2*x2 + b3*x3y1 = b0*x1 + b1*x2 + b3*x4+ b2*x3y2 = b0*x2 + b1*x3 + b3*x5+ b2*x4


time

y[n]

0 1 2

*B5*B5*B5*B5

*A5*A5*A5*A5

b0b1b2b3


FIR Code

Code location: Code\Chapter 14 - Finite Impulse Response Filters

Projects: Fixed Point in C: \FIR_C_Fixed\ Floating Point in C: \FIR_C_Float\ Fixed Point in Assembly: \FIR_Asm_Fixed\ Floating Point in Assembly: \FIR_Asm_Float\

WEEK 8Finite Impulse Response (FIR) Filters

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EEM478-WEEK8 Finite Impulse Response (FIR) Filters 478/icerik/eem478... · 2018. 4. 11. · Chapter 14, Slide 3 Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2002 Introduction